Douglas Hofstadter - 03 Metamagical Themas Chpater 03 - On Viral Sentences and Self-Replicating Structures

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On Viral Sentences andSelf-Replicating Structures

January, 1983

TWO years ago, when I first wrote about self-referential sentences, I was hit by an

avalanche of mail from readers intrigued by the phenomenon of self-reference in its many different guises. I had the chance to print some of those responses one year ago, and that column then triggered a second wave of replies. Many of them have cast self-reference in new light of various sorts. In this column, I would like to describe the ideas of several people, two of whom responded to my initial column with remarkably similar letters: Stephen Walton of New York City and Donald R. Going of Oxon Hill, Maryland.

Walton and Going saw self-replicating sentences as similar to virusessmall objects that enslave larger and more self-sufficient "host" objects, getting the hosts by hook or by crook to carry out a complex sequence of replicating operations that bring new copies into being, which are then free to go off and enslave further hosts, and so on. "Viral sentences", as Walton called them, are "those that seek to obtain their own reproduction by commandeering the facilities of more complex entities".

Both Walton and Going were struck by the perniciousness of such sentences: the selfish way in which they invade a space of ideas and, merely by making copies of themselves all over the place, manage to take over a large portion of that space. Why do they not manage to overrun all of that idea-space? A good question. The answer should be obvious to students of evolution: competition from other self-replicators. One type of replicator seizes a region of the space and becomes good at fending off rivals; thus a "niche" in idea-space is carved out.

This idea of an evolutionary struggle for survival by self-replicating ideas is not original with Walton or Going, although both had fresh things to say on it. The first reference I know of to this notion is in a passage by neurophysiologist Roger Sperry in an article he wrote in 1965 called "Mind, Brain, and Humanist Values". He says: "Ideas cause ideas and help evolve

On Viral Sentences and Self-Replicating Structures 49

new ideas. They interact with each other and with other mental forces in the same brain, in neighboring brains, and, thanks to global communication, in far distant, foreign brains. And they also interact with the external surroundings to produce in toto a burstwise advance in evolution that is far beyond anything to hit the evolutionary scene yet, including the emergence -of the living cell."

Shortly thereafter, in 1970, the molecular biologist Jacques Monod came out with his richly stimulating and provocative, book Chance and Necessity. In its last chapter, "The Kingdom and the Darkness", he wrote of the selection of ideas as follows:

For a biologist it is tempting to draw a parallel between the evolution of ideas and that of the biosphere. For while the abstract kingdom stands at a yet greater distance above the biosphere than the latter does above the nonliving universe, ideas have retained some of the properties of organisms. Like them, they tend to perpetuate their structure and to breed; they too can fuse, recombine, segregate their content; indeed they too can evolve, and in this evolution selection must surely play an important role. I shall not hazard a theory of the selection of ideas. But one may at least try to define some of the principal factors involved in it. This selection must necessarily operate at two levels: that of the mind itself and that of performance.

The performance value of an idea depends upon the change it brings to the behavior of the person or the group that adopts it. The human group upon which a given idea confers greater cohesiveness, greater ambition, and greater self-confidence thereby receives from it an added power to expand which will insure the promotion of the idea itself. Its capacity to "take", the extent to which it can be "put over" has little to do with the amount of objective truth the idea may contain. The important thing about the stout armature a religious ideology constitutes for a society is not what goes into its structure, but the fact that this structure is accepted, that it gains sway. So one cannot well separate such an idea's power to spread from its power to perform.

The "spreading power"-the infectivity, as it were-of ideas, is much more difficult to analyze. Let us say that it depends upon preexisting structures in the mind, among them ideas already implanted by culture, but also undoubtedly upon certain innate structures which we are hard put to identify. What is very plain, however, is that the ideas having the highest invading potential are those that explain man by assigning him his place in an immanent destiny, in whose bosom his anxiety dissolves.

Monod refers to the universe of ideas, or what I earlier termed "idea-space", as "the abstract kingdom". Since he portrays it as a close analogue to the biosphere, we could as well call it the "ideosphere".

* * *

In 1976, evolutionary biologist Richard Dawkins published his book The Selfish Gene, whose last chapter develops this theme further. Dawkins' name


for the unit of replication and selection in the ideosphere--the ideosphere's counterpart to the biosphere's gene-is meme, rhyming with "theme" or "scheme". As a library is an organized collection of books, so a memory is an organized collection of memes. And the soup in which memes grow and flourish-the analogue to the "primordial soup" out of which life first oozed-is the soup of human culture. Dawkins writes:

Examples of memes are tunes, ideas, catch-phrases, clothes fashions, ways of making pots or of building arches. Just as genes propagate themselves in the gene pool by leaping from body to body via sperms or eggs, so memes propagate themselves in the meme pool by leaping from brain to brain via a process which, in the broad sense, can be called imitation. If a scientist hears, or reads about, a good idea, he passes it on to his colleagues and students. He mentions it in his articles and his lectures. If the idea catches on, it can be said to propagate itself, spreading from brain to brain. As my colleague N. K. Humphrey neatly summed up an earlier draft of this chapter: ' . . . memes should be regarded as living structures, not just metaphorically but technically. When you plant a fertile meme in my mind you literally parasitize my brain, turning it into a vehicle for the meme's propagation in just the way that a virus may parasitize the genetic mechanism of a host cell. And this isn't just a way of talking-the meme for, say, `belief in life after death' is actually realized physically, millions of times over, as a structure in the nervous systems of individual men the world over.'

Consider the idea of God. We do not know how it arose in the meme pool. Probably it originated many times by independent `mutation'. In any case, it is very old indeed. How does it replicate itself? By the spoken and written word, aided by great music and great art. Why does it have such high survival value? Remember that `survival value' here does not mean value for a gene in a gene pool, but value for a meme in a meme pool. The question really means: What is it about the idea of a god which gives it its stability and penetrance in the cultural environment? The survival value of the god meme in the meme pool results from its great psychological appeal. It provides a superficially plausible answer to deep and troubling questions about existence. It suggests that injustices in this world may be rectified in the next. The 'everlasting arms' hold out a cushion against our own inadequacies which, like a doctor's placebo, is none the less effective for being imaginary. These are some of the reasons why the idea of God is copied so readily by successive generations of individual brains. God exists, if only in the form of a meme with high survival value, or infective power, in the environment provided by human culture.

Dawkins takes care here to emphasize that there need not be an exact copy of each meme, written in some universal memetic code, in each person's brain. Memes, like genes, are susceptible to variation or distortion-the analogue to mutation. Various mutations of a meme will have to compete with each other, as well as with other memes, for attention-which is to say, for brain resources in terms of both space and time devoted to that meme. Not only must memes compete for inner resources, but, since they are


transmissible visually and aurally, they must 'compete for radio and television time, billboard space, newspaper and magazine column-inches, and library shelf-space. Furthermore, some memes will tend to discredit others, while some groups of memes will tend to be internally self-reinforcing. Dawkins says:

... Mutually suitable teeth, claws, guts, and sense organs evolved in carnivore gene pools, while a different stable set of characteristics emerged from herbivore gene pools. Does anything analogous occur in meme pools? Has the god meme, say, become associated with any other particular memes, and does this association assist the survival of each of the participating memes? Perhaps we could regard an organized church, with its architecture, rituals, laws, music, art, and written tradition, as a co-adapted stable set of mutually-assisting memes.

To take a particular example, an aspect of doctrine which has been very effective in enforcing religious observance is the threat of hell fire. Many children and even some adults believe that they will suffer ghastly torments after death if they do not obey the priestly rules. This is a particularly nasty technique of persuasion, causing great psychological anguish throughout the middle ages and even today. But it is highly effective. It might almost have been planned deliberately by a machiavellian priesthood trained in deep psychological indoctrination techniques. However, I doubt if the priests were that clever. Much more probably, unconscious memes have ensured their own survival value by virtue of those same qualities of pseudo-ruthlessness which successful genes display. The idea of hell fire is, quite simply, self-perpetuating, because of its own deep psychological impact. It has become linked with the god meme because the two reinforce each other, and assist each other's survival in the meme pool.

Another member of the religious meme complex is called faith. It means blind trust, in the absence of evidence, even in the teeth of evidence .... Nothing is more lethal for certain kinds of meme than a tendency to look for evidence .... The meme for blind faith secures its own perpetuation by the simple unconscious expedient of discouraging rational inquiry.

Blind faith can justify anything. If a man believes in a different god, or even if he uses a different ritual for worshipping the same god, blind faith can decree that he should die-on the cross, at the stake, skewered on a Crusader's sword, shot in a Beirut street, or blown up in a bar in Belfast. Memes for blind faith have their own ruthless ways of propagating themselves. This is true of patriotic and political as well as religious blind faith.

* * *

When I muse about memes, I often find myself picturing an ephemeral flickering pattern of sparks leaping from brain to brain, screaming "Me, me!" Walton's and Going's letters reinforced this image in interesting ways. For instance, Walton begins with the simplest imaginable viral sentences "Say me!" and "Copy me! "-and moves quickly to more complex variations with blandishments ("If you copy me, I'll grant you three wishes!") or


threats ("Say me or I'll put a curse on you!"), neither of which, he observes, is likely to be able to keep its word. Of course, as he points out, this may not matter, the only final test of viability being success at survival in the meme pool. All's fair in love and war-and war includes the eternal battle for survival, in the ideosphere no less than in the biosphere.

To be sure, very few people above the age of five will fall for the simple-minded threats or promises of these sentences. However, if you simply tack on the phrase "in the afterlife", far more people will be lured into the memetic trap. Walton observes that a similar gimmick is used by your typical chain letter (or "viral text"), which "promises wealth to those who faithfully replicate it and threatens doom to any who fail to copy it". Do you remember the first time you received such a chain letter? Do you recall the sad tale of "Don Elliot, who received $50,000 but then lost it because he broke the chain"? And the grim tale of "General Welch in the Philippines, who lost his life [or was it his wife?] six days after he received this letter because he failed to circulate the prayer-but before he died, he received $775,000"? Poor Don Elliot! Poor General Welch! It's hard not to be just a little sucked in by such tales, even if you wind up throwing the letter out contemptuously.

I found Walton's phrases "viral sentence" and "viral text" to be exceedingly catchy-little memes in themselves, definitely worthy of replication some 700,000 times in print, and who knows how many times orally beyond that. At least that's my opinion. Of course, it also depends on how the editor of Scientific American feels. [It turned out he felt fine about it.] Well, now, Walton's own viral text, as you can see here before your eyes, has managed to commandeer the facilities of a very powerful host-an entire magazine and printing press and distribution service. It has leapt aboard and is now-even as you read this viral sentence-propagating itself madly throughout the ideosphere!

This idea of choosing the right host is itself an important aspect of the quality of a viral entity. Walton puts it this way:

The recipient of a viral text can, of course, make a big difference. A tobacco mosaic virus that attacks a salt crystal is out of luck, and some people rip up chain letters on sight. A manuscript sent to an editor may be considered viral, even though it contains no explicit self-reference, because it is attempting to secure its own reproduction through an appropriate host; the same manuscript sent to someone who has nothing to do with publishing may have no viral quality at all.

As it concludes, Walton's letter graciously steps forward from the page and squeaks to me directly on its own behalf: "Finally, I (this text) would be delighted to be included, in whole or in part, in your next discussion of self-reference. With that in mind, please allow me to apologize in advance for infecting you."

* * *


Whereas Walton mentioned Dawkins in his letter, Going seems not to have been aware of Dawkins at all, which makes his letter quite remarkable in its close connection to Dawkins' ideas. Going suggests that we consider, to begin with, Sentence A:

It is your duty to convince others that this sentence is true.

As he says:

If you were foolish enough to believe this sentence, you would attempt to convince your friends that A is true. If they were equally foolish, they would convince their friends, and so on until every human mind contained a copy of A. Thus, A is a self-replicating sentence. More particularly, it is the intellectual equivalent of a virus. If Sentence A were to enter a mind, it would take control of the mind's intellectual machinery and use it to produce hundreds of copies of itself in other minds.

The problem with Sentence A, of course, is that it is absurd; no one could possibly believe it. However, consider the following:

System S:Begin:

S1: Blah.S2: Blah blah.S3: Blah blah blah.. .. .. .. .S99: Blah blah blah blah blah blab S100: It is your duty to convince others that System S is true.

End.

Here, S1 through S99 are meant to be statements that constitute a belief system having some degree of coherency. If System S taken as a whole were convincing, then the entire system would be self-replicating. System S would be especially convincing if 5100 were not stated explicitly but held as a logical consequence of the other ideas in the system.

Let us refer to Going's S100 as the hook of System S, for it is by this hook that System S hopes to hoist itself onto a higher level of power. Note that on its own, a hook that in effect says "It is your duty to believe me" is not a viable viral entity; in order to "fly", it needs to drag something extra along with it, just as a kite needs a tail to stabilize it. Pure lift goes out of control and self-destructs, but controlled lift can lift itself along with its controller. Similarly, 5100 and SI-S99 (taken as a set) are symbiotes: they play


complementary, mutually supportive roles in the survival of the meme they together constitute. Now Going develops this theme a little further:

Statements S,-S99 are the bait which attracts the fish and conceals the hook. No bait-no bite. If the fish is fool enough to swallow the baited hook, it will have little enough time to enjoy the bait. Once the hook takes hold, the fish will lose all its fishiness and become instead a busy factory for the manufacture of baited hooks.

Are there any real idea systems that behave like System S? I know of at least three. Consider the following:

System X:Begin:

X1: Anyone who does not believe System X will burn in hell, X2: It is your duty to save others from suffering.

End.

If you believed in System X, you would attempt to save others from hell by convincing them that System X is true. Thus System X has an implicit `hook' that follows from its two explicit sentences, and so System X is a self-replicating idea system. Without being impious, one may suggest that this mechanism has played some small role in the spread of Christianity.Self-replicating ideas are most often found in politics. Consider Sentence IV.

The whales are in danger of extinction.

If you believed this idea, you would want to save the whales. You would quickly discover that you could not reach this goal by yourself. You would need the help of thousands of like-minded people. The first step in getting their help would be to convince them that Sentence i1' is true. Thus a'hook' like 5100 follows from Sentence II, and Sentence IV is a self-replicating idea.In a democracy, nearly any idea will tend to replicate since the only way to win an election is to convince other people to share your ideas. Most political ideas are not properly self-replicating, since the motive for spreading the idea is separate from the idea itself. Statement IV, on the other hand, is genuinely self-replicating, since the duty to propagate it is a direct logical consequence of IV itself. Ideas like W can sometimes take on a life of their own and drive their own propagation.A more sinister form of self-replication is Sentence B:

The bourgeoisie is oppressing the proletariat.

This statement is self-replicating for the same reason as W is. The desire to propagate statements like B is driven by a desire to protect a victim figure from a villain figure. Such ideas are dangerous because belief in them may lead to attacks on the supposed villain. Statement B also illustrates the fact


that the self-replicating character of an idea depends only upon the idea's logical structure, not upon its truth.

Statement B is merely a special case of the generalized statement, Sentence V:

The villain is wronging the victim.

Here, the word villain must be replaced with the name of some real group (capitalists, communists, imperialists, Jews, freemasons, aristocrats, men, foreigners, etc.). Likewise, victim must be replaced with the name of the corresponding victim and wronging filled in as desired. The result will be a self-replicating idea system for the same reasons as lV and B were. Note that each of the suggested substitutions yields a historically attested idea system. It has long been recognized that most extremist mass movements are based on a belief similar to V. Part of the reason seems to be that type-l' statements reduce to the 'hook', S100, and therefore define self-replicating idea systems. One hesitates to explain real historical events in terms of such a silly mechanism, and yet ....

Going brings his ideas to an amusing conclusion as follows:

Suppose we parody my thesis by proposing Sentence E:

The self-replicating ideas are conspiring to enslave our minds.

This 'paranoid' statement is clearly an idea of type l'. Thus, the thesis seems to describe itself. Further, if we accept E, then we must say that this type-V idea implies that we must distrust all ideas of type P. This is the Epimenides Paradox.

It is interesting that all these people who have explored these ideas have given examples ranging from the very small scale of such things as catchy tunes (for example, Dawkins cites the opening theme of Beethoven's fifth symphony) and phrases (the word "meme" itself) to the very large scale of ideologies and religions. Dawkins uses the term meme complex for these larger agglomerations of memes; however, I prefer the single word scheme.

One reason I prefer it is that it fits so well with the usage suggested by psychiatrist and writer Allen Wheelis in his novel The Scheme of Things. Its central character is a psychiatrist and writer named Oliver Thompson, whose darkly brooding essays are scattered throughout the book, interspersed with brightly colored, evocative episodes. Thompson is obsessed with the difference between, on the one hand, "the raw nature of existence, unadorned, unmediated", which he refers to repeatedly as "the way things are", and, on the other hand, "schemes of things", invented by ' humans-ways of making order and sense out of the way things are. Here are some of Thompson's musings on that theme:


I want to write a book .... the story of one man whose life becomes a metaphor for the entire experience of man on earth. It will portray his search through a succession of schemes of things, show the breakdown, one after another, of each pattern he finds, his going on always to another, always in the hope that the scheme of things he finds and for the moment is serving is not a scheme of things at all but reality, the way things are, therefore an absolute that will endure forever, within which he can serve, to which he can contribute, and through, which he can give his mortal life meaning and so achieve eternal life....

The scheme of things is a system of order. Beginning as our view of the world, it finally becomes our world. We live within the space defined by its coordinates. It is self-evidently true, is accepted so naturally and automatically that one is not aware of an act of acceptance having taken place. It comes with one's mother's milk, is chanted in school, proclaimed from the White House, insinuated by television, validated at Harvard. Like the air we breathe, the scheme of things disappears, becomes simply reality, the way things are. It is the lie necessary to life. The world as it exists beyond that scheme becomes vague, irrelevant, largely unperceived, finally nonexistent ....

No scheme of things has ever been both coextensive with the way things are and also true to the way things are. All schemes of things involve limitation and denial ....

A scheme of things is a plan for salvation. How well it works will depend upon its scope and authority. If it is small, even great achievement in its service does little to dispel death. A scheme of things may be as large as Christianity or as small as the Alameda County Bowling League. We seek the largest possible scheme of things, not in a reaching out for truth, but because the more comprehensive the scheme the greater its promise of banishing dread. If we can make our lives mean something in a cosmic scheme we will live in the certainty of immortality. Those attributes of a scheme of things that determine its durability and success are its scope, the opportunity it offers for participation and contribution, and the conviction with which it is held as self-evidently true. The very great success of Christianity for a thousand years follows upon its having been of universal scope, including and accounting for everything, assigning to all things a proper place; offering to every man, whether prince or beggar, savant or fool, the privilege of working in the Lord's vineyard; and being accepted as true throughout the Western world.

As a scheme of things is modified by inroads from outlying existence, it loses authority, is less able to banish dread; its adherents fall away. Eventually it fades, exists only in history, becomes quaint or primitive, becomes, finally, a myth. What we know as legends were once blueprints of reality. The Church was right to, stop Galileo; activities such as his import into the regnant scheme of things new being which will eventually destroy that scheme.

Taken in Wheelis' way, "scheme" seems a fitting replacement for Dawkins' "meme complex". A scheme imposes a top-down kind of perceptual order on the world, propagating itself ruthlessly, like Going's System S with its "hook". Wheelis' description of the inadequacy of all "schemes of things" to fully and accurately capture "the way things are" is strongly reminiscent of the vulnerability of all sufficiently powerful formal


systems to either incompleteness or inconsistency-a vulnerability that ensues from another kind of "hook": the famous Gödelian hook, which arises from the capacity for self-reference of such systems, although neither Wheelis nor Thompson makes any mention of the analogy. We shall come back to Gödel momentarily.

* * *

The reader of this novel must be struck by the professional similarity of Wheelis and his protagonist. It is impossible to read the book and not to surmise that Thompson's views are reflecting Wheelis' own views-and yet, who can say? It is a tease. Even more tantalizing is the title of Thompson's imaginary book, which Wheelis casually mentions toward the end of the novel: it is The Way Things Are-a striking contrast to the title of the real book in which it exists. One wonders: What is the meaning of this elegant literary pleat in which one level folds back on another? What is the symbolism of Wheelis within Wheelis?

Such a twist, by which a thing (sentence, book, system, person) seems to refer to itself but does so only by allusion to something resembling itself, is called indirect self-reference. You can do this by pointing at your image in a mirror and saying, "That person sure is good-looking!" That one is very simple, because the connection between something and its mirror image is so familiar and obvious-seeming to us that there seems to be no distance whatsoever between direct and indirect referents: we equate them completely. Thus it seems there is no referential indirectness.

On the other hand, this depends upon the ease with which our perceptual systems convert a mirror image into its reverse, and upon other qualities of our cognitive systems that allow us to see through several layers of translation without being aware of the layers-like looking through many feet of water and seeing not the water but only what lies at its bottom.

Some indirect self-references are of course subtler than others. Consider the case of Matt and Libby, a couple ostensibly having a conversation about their friends Tammy and Bill. It happens that Matt and Libby are having some problems in their relationship, and those problems are quite analogous to those of Tammy and Bill, only with sexes reversed: Matt is to Libby what Tammy is to Bill, in their respective relationships. So as Matt and Libby's conversation progresses, although on the surface level it is completely about their friends Tammy and Bill, on another level it is actually about themselves, as reflected in these other people. It is almost as if, by talking about Tammy and Bill, Matt and Libby are going over a fable by Aesop that has obvious relevance to their own plight. There are things going on simultaneously on two levels, and it is hard to tell how conscious either of the participants is of the exchange of dual messages-one of concern about their friends, one of concern about themselves.


Indirect self-reference can be exploited in the most unexpected and serious ways. Consider the case of President Reagan, who on a recent occasion of high Soviet-American tension over Iran, went out of his way to recall President Truman's behavior in 1945, when Truman made some very blunt threats to the Soviets about the possibility of the U.S. using nuclear weapons if need be against any Soviet threat in Iran. Merely by bringing up the memory of that occasion, Reagan was inviting a mapping to be made between himself and Truman, and thereby he was issuing a not-so-veiled threat, though no one could point to anything explicit. There simply was no way that a conscious being could fail to make the connection. The resemblance of the two situations was too blatant.

Thus, does self-reference really come in two varieties-direct and indirect -or are the two types just distant points on a continuum? I would say unhesitatingly that it is the latter. And furthermore, you can delete the prefix "self ", so that the question becomes one of reference in general. The essence is simply that one thing refers to another whenever, to a conscious being, there is a sufficiently compelling mapping between the roles the two things are perceived to play in some larger structures or systems. (See Chapter 24 for further discussion of the perception of such roles.) Caution is needed here. By "conscious being", I mean an analogy-hungry perceiving machine that gets along in the world thanks to its perceptions; it need not be human or even organic. Actually, I would carry the abstraction of the term "reference" even further, as follows. The mapping of systems and roles that establishes reference need not actually be perceived by any such being: it suffices that the mapping exist and simply be perceptible to such a being were it to chance by.

* * *

The movie The French Lieutenant's Woman (based on John Fowles' novel of the same name) provides an elegant example of ambiguous degrees of reference. It consists of interlaced vignettes from two concurrently developing stories both of which involve complex romances; one takes place in Victorian England, the other in the present. The fact that there are two romances already suggests, even if only slightly, that a mapping is called for. But much more is suggested than that. There are structural similarities between the two romances: each of them has triangular qualities, and in both stories, only one leg of the triangle is focused upon. Moreover, the same two actors play the two lovers in both romances, so that you see them in alternating contexts and with alternating personality traits. The reason for this "coincidence" is that the contemporary story concerns the making of a film of the Victorian story.

As the two stories unfold in parallel, a number of coincidences arise that suggest ever more strongly that a mapping should be made. But it is left to the movie viewer to carry this mapping out; it is never called for explicitly.


After a time, though, it simply becomes unavoidable. What is pleasant in this game is the fluidity left to the viewer: there is much room for artistic license in seeing connections, or suspecting or even inventing connections.

Indirect reference of the artistic type is much less precise than indirect reference of the formal type. The latter arises when two formal systems are isomorphic-that is, they have strictly analogous internal structures, so that there is a rigorous one-to-one mapping between the roles in the one and the roles in the other. In such a case, the existence of genuine reference becomes as clear to us as in the case of someone talking about their mirror image: we take it as immediate, pure self-reference, without even noticing the indirectness, the translational steps mediated by the isomorphism. In fact, the connection may seem too direct even to be called "reference"; some may see it simply as identity.

This perceptual immediacy is the reason that Gödel’s famous sentence G of mathematical logic is said to be self-referential. Everyone accepts the idea that G talks about a number, g (though a radical skeptic might question even that!); the tricky Gödelian step is in seeing that g (the number) plays a role in the system of natural numbers strictly analogous to the role that G (the sentence) plays in the axiomatic system it is expressed in. This Wheelis-like oblique reference by G to itself via its "image" g is generally accepted as genuine self-reference. (Note that we have even one further mapping: G plays the role of Wheelis, and its Gödel number g that of Wheelis' alter ego Thompson.)

The two abstract mappings that, when telescoped, establish G's self-reference but make it seem indirect can be collapsed into just one mapping, following a slogan that we might formulate this way: "If A refers to B, and B is just like C, then A refers to C." For instance, we can let A and C be Wheelis, with B being Thompson. This makes Wheelis' self-reference a "theorem". Of course, this "theorem" is not rigorously proven, since our slogan has to be taken with a grain of salt. Being `just like" something else is a highly disputable matter.

However, in a formal context where is jurt like is virtually synonymous with plays a role isomorphic to that of, then the slogan can have a strict meaning, and thereby justify a theorem more rigorously. In particular, if A and C are equated with G, and B with g, then our slogan runs: "If G refers to g, and g plays a role isomorphic to that of G, then G refers to G." Since the premises are true, the conclusion must be true. According to this scheme of things, then, G is a genuinely self-referential sentence, rather than some sort of logical illusion as deceptive as an Escher print.

* * *

Indirect self-reference suggests the idea of indirect self-replication, in which a viral entity, instead of replicating itself exactly, brings into being another entity that plays the same role as it does, but in some other system: perhaps


its mirror image, perhaps its translation into French, perhaps a string of the product numbers of all its parts, together with pre-addressed envelopes containing checks made out to the factories where those parts are made, and a list of instructions telling what to do with all the parts when they arrive in the in :l.

This may sound familiar to some readers. In fact, it is an indirect reference to the Von Neumann Challenge, the puzzle posed in Chapter 2 to create a self-describing sentence whose only quoted matter is at the word or letter level, rather than at the level of whole quoted phrases. I discovered, as I received candidate solutions, that many readers did not understand what this requirement meant. The challenge came out of an objection to the complexity of the "seed" (the quoted part) in Quine's version of the Epimenides paradox:

"yields falsehood when appended to its quotation." yields falsehood when appended to its quotation.

To see what is strange here, imagine that you wish to have a space-roving robot build a copy of itself out of raw materials that it encounters in its travels. Here is one way you could do it: Make the robot symmetrical, like a human being. Also make the robot able to make a mirror-image copy of any structure that it encounters along its way. Finally, have the robot be programmed to scan the world constantly, the way a hawk scans the ground for rodents. The search image in the robot's case is that of an object identical to its own left half. The robot need not be aware that its target is identical to its left half; the search can go on merrily for what seems to it to be merely a very complex and arbitrary structure. When, after scouring the universe for seventeen googolplex years, it finally comes across such a structure, then of course the robot activates its mirror-image-production facility and creates a right half. The last step is to fasten the two halves together, and presto! A copy emerges. Easy as pie-provided you're willing to wait seventeen googolplex years (give or take a few minutes).

The arbitrary and peculiar aspect of the Quine sentence, then, is that its seed is half as complex-which is to say, nearly as complex-as the sentence itself. If we resume our robot parable, what we'd ideally like in a self-replicating robot is the ability to make itself literally from the ground up: let us say, for instance, to mine iron ore, to smelt it, to cast it in molds to make nuts and bolts and sheet metal and so on; and finally, to be able to assemble the small parts into larger and larger subunits until, miraculously, a replica is born out of truly raw materials. This was the spirit of the Von Neumann Challenge: I wanted a linguistic counterpart to this "self-replicating robot of the second kind".

In particular, this means a self-documenting or self-building sentence that builds both its halves-its quoted seed and its unquoted building rule-out of linguistic raw materials (words or letters). Many readers failed to


understand what this implies. The most common mistake was to present, as the seed, a long sequence of individually quoted words (or letters) in a specific order, then to exploit that order in the building rule. Well then, you might as well have quoted one big long ordered string, as Quine did. The idea of my challenge was that all structure in the built object must arise exclusively out of some principle enunciated in the building rule, not out of the seed's internal structure.

Just as a self-replicating robot in some random alien environment is hardly likely to find all its parts lined up on a shelf in order of assembly but must rely on its "brain" or program to recognize raw parts wherever and whenever they turn up so that it can grab them and therefrom assemble a copy of itself, so the desired sentence must treat the pieces of the seed without regard to the order in which they are listed, yet must be able to construct itself in the proper order out of them. Thus it's fine if you enclose the entire seed within a single pair of quotes, rather than quoting each word individually-all that matters is that the seed's word order (or better yet, its letter order) not be exploited. The seed of the ideal solution would be a long inventory of parts, similar to the list of ingredients of a recipe-perhaps a list of 50 'e's, then 46 Ts, and so on. Clearly those letters cannot remain in that order; they simply constitute the raw materials out of which the new sentence is to be built.

* * *

Nobody sent in a solution whose seed was at the primordial level of letters. A few people, however, did send in adequate, if not wonderfully elegant, solutions with seeds at the word level. The first correct solution I received came from Frank Palmer of Chicago, who therefore receives the first 'Johnnie" award-a self-replicating dollar bill given to the Grand Winner of the First Every-Other-Decade Von Neumann Challenge. Unfortunately, the dollar bill consumes the entire body of its owner in its bizarre process of self-replication, and so it is wisest to simply lock it up to protect oneself from its voracious appetite.

Palmer submitted several versions. In them, he utilized upper and lower cases to distinguish between seed and building rule, respectively. Here is one solution, slightly modified by me:

after alphabetizing, decapitalize FOR AFTER WORDS STRING FINALLY UNORDERED UPPERCASE FGPBVKXQ/Z NONVOCALIC DECAPITALIZE SUBSTITUTING ALPHABETIZING, finally for nonvocalic string substituting unordered uppercase words

Let us watch how it works, step by careful step. We must bear in mind that the instructions we are following are the lowercase words printed above, and that the uppercase words are not to be read as instructions. Nor, for that


matter, are the lowercase words that we will soon be working with. They are like the inert, anesthetized body of a patient being operated on, who, when the operation is over, will awake and become animate. So let's go. First we are to alphabetize the seed. (I am treating the comma as attached to the word preceding it.) This gives us the following:

AFTER ALPHABETIZING, DECAPITALIZE FGPBVKXQJZ FINALLY FOR NONVOCALIC STRING SUBSTITUTING UNORDERED UPPERCASE WORDS

Next we are to decapitalize it. This will yield some lowercase words-the "anesthetized" lowercase words I spoke of above:

after alphabetizing, decapitalize fgpbvkxqjz finally for nonvocalic string substituting unordered uppercase words

All right; now our final instruction is to locate a nonvocalic string (that's easy: ` fgpbvkxgjz ") and to substitute for it the uppercase words, in any order (that is, the original seed itself, but without regard for its structure above the level of the individual word-unit). This last bit of surgery yields:

after alphabetizing, decapitalize SUBSTITUTING FINALLY WORDS UNORDERED STRING DECAPITALIZE UPPERCASE FOR NONVOCALIC AFTER FGPBVKXQJZ ALPHABETIZING, finally for nonvocalic string substituting unordered uppercase words

And this is a perfect copy of our starting sentence! Or rather, semiperfect. Why only semiperfect? Because the seed has been randomly scrambled in the act of self-reproduction. The beauty of the scheme, though, is that the internal structure of the seed is entirely irrelevant to the efficacy of the sentence as a self-replicator. All that matters is that the new building rule say the proper thing, and it will do so no matter what order the seed from which it sprang was in. Now this fresh new baby sentence can wake up from its anesthesia and go off to replicate itself in turn.

The critical step was the first one: alphabetization. This turns the arbitrarily-ordered seed into a grammatical, meaningful command-merely by mechanically exploiting a presumed knowledge of the "ABC"s. But why not? It is perfectly reasonable to presume superficial typographical knowledge about letters and words, since such knowledge deals with printed material as raw material: purely syntactically, without regard to the meanings carried therein. This is just like the way that enzymes in the living cell deal with the DNA and RNA they chop up and alter and piece together again: purely chemically, without regard to the "meanings" carried therein. Just as chemical valences and affinities and so on are taken as givens in the workings


of the cell, so alphabetic and typographic facts are taken as givens in the V. N. Challenge.When Palmer sent in his solution, he happened to write down his seed in order of

increasing length of words,- but that is inessential; any random order would have done, and that sort of idea is the crucial point that many readers missed. Another rather elegant solution was sent in by Martin Weichert of Munich. It runs this way (slightly modified by me):

Alphabetize and append, copied in quotes, these words: "these append, in Alphabetize and words: quotes, copied"

It works on the same principle as Palmer's sentence, and again features a seed whose internal structure (at least at the word level) is irrelevant to successful self-replication. Weichert also sent along an intriguing palindromic solution in Esperanto, in which the flexible word order of the language plays a key role. Michael Borowitz and Bob Stein of Durham, North Carolina sent in a solution similar to Palmer's.

* * *

Finally, last year's gold-medal winner for self-documentation, Lee Sallows, was a bit piqued by my suggestion that the gold on his medal was somewhat tarnished since he had not paid close enough attention to the use-mention distinction. Apparently I goaded him into constructing an even more elaborate self-documenting sentence. Although it does not quite fit what I had in mind for the Von Neumann Challenge, as it does not spell out its own construction explicitly at the letter level or word level, it is another marvelous Sallowsian gem, and I shall therefore generously allow the gold on his medal to go untarnished this year. (Apologies to those purists who insist that gold doesn't tarnish. I must have been confusing it with copper and silver. How silly of me!) Herewith follows Sallows' 1982 contribution:


*Write

down ten 'a 's,eight 'c's, ten Vs,

fifty-two 'e's, thirty-eight fs,sixteen g's, thirty 'h 's, forty-eight 'i's,

six 'l's, four 'm's, thirty-two `n's, forty-four 'o's,four Ps, four 'q's, forty-two 'r's, eighty-four 's's,

seventy-six 't's, twenty-eight 'us, four 'v s, four 'W's,eighteen 'w's, fourteen 'x's, thirty-two y's, four ':s,

four '*'s, twenty-six '-'s, fifty-eight ', s,sixty "'s and sixty "'s, in a

palindromic sequencewhose second

half runsthus::suht

snur Jahdnoces esohw

ecneuqes cimordnilapa ni s "' ytxis dna s "' ytxis

,S',' thgie-ytf,s'-' xis-ytnewt ,s'*' ruof,s':' ruof,s y' owt-ytriht ,s'x' neetruof,s'w' neethgie

,s'W' roof s 'v' ruof ,s'u' thgie-ytnewt ,s't' xis-ytnevess's' ruof-ythgie ,s 'r' owt-ytrof ,s 'q' ruof ,s p' ruof,s 'o' ruof-ytrof ,s'n' owl-ytriht ,s 'm' ruof s 'l' xis

,s'i' thgie-ytrof ,s'h' ytriht s g' neetxis,s f thgie-ytriht ,s'e' owt-ytfif

,s d' net ,s'c' thgies' a' net nwod

etirW*

Post Scriptum

After writing this column, I received much mail testifying to the fact that there are a large number of people who have been infected by the "meme" meme. Arel Lucas suggested that the discipline that studies memes and their connections to humans and other potential carriers of them be known as memetics, by analogy with "genetics". I think this is a good suggestion, and hope it will be adopted.

Maurice Gueron wrote me from Paris to tell me that he believed the first clear exposition of the idea of self-reproducing ideas that inhabit the brains


of organisms was put forward in 1952 by Pierre Auger, a physicist at the Sorbonne, in his book L'homme microscopique. Cueron sent me a photocopy of the relevant portions, and I could indeed see how prophetic the book was.

I received a copy of the book General Theory of Evolution by Vilmos Csdnyi, a Hungarian geneticist. In this book, he attempts to work out a theory in which memes and genes evolve in parallel. A similar attempt is made in the book Ever-Expanding Horizons: The Dual Informational Sources of Human Evolution, by the American biologist Carl B. Swanson.

The most thorough-going research on the topic of pure memetics I have yet run across is that of Aaron Lynch, an engineering physicist at Fermilab in Illinois, who in his spare time is writing a book called Abstract Evolution. The portions that I have read go very carefully into the many "options", to speak anthropomorphically, that are open to a meme for getting itself reproduced over and over in the ideosphere (a term Lynch and I invented independently). It promises to be a provocative book, and I look forward to its publication.

* * *

Jay Hook, a mathematics graduate student, was provoked by the solutions to the Von Neumann Challenge as follows:

The notion that it takes two to reproduce is suggestive. Perhaps a change in terminology is appropriate. The component that you call the "seed" might be thought of as the "female" fragment-the egg that grows into an adult, but only after receiving instructions from the sperm, the "male" fragment-the building rule. In this interpretation, our sentences say everything twice because they are hermaphroditic: the male and female fragments appear together in the same individual.

To better mimic nature, we should construct pairs of sentences or phrases, one male and one female-expressions that taken individually produce nothing but when put together in a dark room make copies of themselves. I propose the following. The male fragment

After alphabetizing and deitalicizing, duplicate female fragment in its original version.

doesn't seem to say much by itself, and the female fragment

in and its After female fragment original version. duplicate alphabetizing deitalicizing,

certainly doesn't, but let them at each other and watch the fireworks. (I follow your practice of assuming each punctuation mark to be attached to the preceding word.) The male takes the lead, and sets to work on the female. First we alphabetize and deitalicize her, he says; that gives a new male fragment.


Then we simply make a copy of her-so we get one of each!Nature still doesn't work this way, of course; it's not clear that couples that

produce offspring only in boy-girl pairs are really superior to self-replicating hermaphrodites. Ideally, our fragments should produce either a copy of the male or a copy of the female, depending on, say, the day of the week or the parity of some external index like the integer part of the current Dow Jones Industrial Average. Surprisingly, this isn't hard. Take the male to be

Alphabetize and deitalicize female fragment if index is odd; otherwise reproduce same verbatim.

and take for the female

if is and odd; same index female fragment otherwise reproduce verbatim. Alphabetize deitalicize

One more refinement. To this point, each offspring has been exactly identical to one of its parents. We can introduce variation, at least in the girls, as follows. Male fragment:

Alphabetize and deitalicize female fragment if index is odd; otherwise randomly rearrange the words.

Female fragment:

if is and the odd,• index female words. fragment randomly otherwise rearrange Alphabetize deitalicize

Now all of the boys will be the spittin' image of their father, but whereas one daughter might be

index rearrange if the Alphabetize randomly fragment odd,- deitalicize is and words. otherwise female

another might be

Alphabetize index and rearrange the fragment if female is odd; otherwise randomly deitalicise words.

The important point, however, is that all of these female offspring, however diverse, are genetically capable of mating with any of the (identical) males. Can you find a way to introduce variation in the males without producing sterile offspring?

In conclusion, allow me to observe that the Dow closed on Friday at 1076.0. Therefore I proudly proclaim: It's a girl!

* * *


I now close by returning to Lee Sallows. This indefatigable researcher of what he calls logological space continued his quest after the holy grail of perfect self-documentation. His jealousy was aroused in the extreme when Rudy Kousbroek, who is Dutch, and Sarah Hart, who is English, together tossed off what Sallows terms "the greatest logological jewel the world has ever seen". Kousbroek and Hart's self-documenting sentence, though in Dutch, ought to be pretty clearly understandable by anyone who takes the time to look at it carefully:

Dit pangram bevat vijf a's, twee b's, twee c's, drie d's, zesenveertig e's, vijf f 's, vier g's, twee h's, vijftien i's, vier j's, een k, twee l's, twee m's, zeventien n's, een o, twee p's, een q, zeven r's, vierentwintig s's, zestien t's, een u, elf v's, acht w's, een x, een y, en zes z's.

In fact, you can learn how to count in Dutch by studying it!There's not an ounce of fat or awkwardness in this sentence, and it drove

Sallows mad that he couldn't come up with an equally perfect pangram (sentence containing every letter of the alphabet) in English. Every attempthad some flaw in it. So in desperation, Sallows, electronics engineer that he is, decided he would design a high-speed dedicated "letter-crunching"machine to search the far reaches of logological space for an equivalent English sentence. Sallows sent me some material on his Pangram Machine.He says:

At the heart of the beast is a clock-driven cascade of sixteen Johnson-counters: the electronic analogue of a stepper-motor-driven stack of combination lock-discs. Every tick of the clock clicks in a new combination of numbers: a unique combination of counter output lines becomes activated .... Pilot tests have been surprisingly encouraging; it looks as though a clock frequency of a million combinations per second is quite realistic. Even so it would take 317 years to explore the ten-deep stratum. But does it have to be ten? With this reduced to a modest but still very worthwhile six-deep range it will take just 32.6 days. Now we're talking!Over the past eight weeks I have devoted every spare second to constructing this rocket for exploring the far regions of logological space .... Will it really fly? So far it looks very promising. And the end is already in sight. With a bit of luck Rudy Kousbroek will be able to launch the machine on its 32-day journey when he comes to visit here at the end of this month. If so, a bottle of champagne will not be out of place.

Two months later, I got a most excited transmission from Lee, which began with the word "EUREKA! "-the word the Pangram Machine was set up to print on success. He then presented three pangrams that his machine had discovered, floating "out there" somewhere beyond the orbit of PlutoMy favorite one is this:


This pangram tallies five a's, one b, one c, two d's, twenty-eight e's, eight fs, six g's, eight h's, thirteen i's, one j, one k, three l's, two m's, eighteen n's, fifteenO's, two p's, one q, seven r's, twenty-five s's, twenty-two. t's, four u's, four v's, nine w's, two X's, four y's, and one z.

Now that's what I call a success for mechanical translation!Sallows writes: "I wager ten guilders that nobody will succeed in producing a

perfect self-documenting solution (or proof of its non-existence) to the sentence beginning, `This computer-generated pangram contains ...'within the next ten years. No tricks allowed. The format to be exactly as in the above pangrams. Either `and' or `&' is permissible. Result to be derived exclusively by von Neumann architecture digital computer (no super computers, no parallel processing). Fancy your chances?" Anyone who wants to write to Sallows can do so, at Buurmansweg 30, 6525 RW Nijmegen, Holland.

Much though I am delighted by Sallows' ingenious machine and his plucky challenge, I expect him to lose his wager before you can say "Raphael Robinson". For my reasons, see the postscript to Chapter 16.


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Douglas Hofstadter - 03 Metamagical Themas Chpater 03 - On Viral Sentences and Self-Replicating Structures